1. Field of the Invention
The present invention relates generally to sensor network and, particularly, to wireless sensor networks in which a plurality of wireless sensors are spread over a geographical location. More particularly, the invention relates to estimation in such a network utilizing a noise-constrained diffusion least mean square method for estimation in the adaptive network.
2. Description of the Related Art
In reference to wireless sensor networks, the term “diffusion” is used to identify the type of cooperation between sensor nodes in the wireless sensor network. That data that is to be shared by any sensor is diffused into the network in order to be captured by its respective neighbors that are involved in cooperation.
Wireless sensor networks include a plurality of wireless sensors spread over a geographic area. The sensors take readings of some specific data and, if they have the capability, perform some signal processing tasks before the data is collected from the sensors for more detailed thorough processing.
A “fusion-center based” wireless network has sensors transmitting all the data to a fixed center, where all the processing takes place. An “ad hoc” network is devoid of such a center and the processing is performed at the sensors themselves, with some cooperation between nearby neighbors of the respective sensor nodes.
Recently, several algorithms have been developed to exploit this nature of the sensor nodes and cooperation schemes have been formalized to improve estimation in sensor networks.
Least mean squares (LMS) algorithms are a class of adaptive filters used to mimic a desired filter by finding the filter coefficients that relate to producing the least mean squares of the error signal (i.e., the difference between the desired and the actual signal). The LMS algorithm is a stochastic gradient descent method, in that the filter is only adapted based on the error at the current time.
FIG. 2 diagrammatically illustrates an adaptive network having N nodes. In the following, boldface letters are used to represent vectors and matrices and non-bolded letters represent scalar quantities. Matrices are represented by capital letters and lower-case letters are used to represent vectors. The notation (.)T stands for transposition for vectors and matrices, and expectation operations are denoted as E[.]. The notation (.)* represents conjugate transposition and, for scalars, (.)* denotes complex conjugation. FIG. 2 illustrates an exemplary adaptive network having N nodes, where the network has a predefined topology. For each node k, the number of neighbors is given by Nk, including the node k itself, as shown in FIG. 2. At each iteration i, the output of the system at each node is given by:dk(i)=uk,iw0+vk.  (1)where uk,i is a known regressor row vector of length M, w0 is an unknown column vector of length M and vk(i) represents additive noise. The output and regressor data are used to produce an estimate of the unknown vector, given by ψk,i.
Diffusion LMS (DLMS) techniques use the data to find an estimate for the unknown vector. There are two strategies that may be employed. In the first strategy, each node combines its own estimate with the estimates of its neighbors using some combination technique, and then the combined estimate is used for updating the node estimate. This method is referred to as Combine-then-Adapt (CIA) diffusion. It is also possible to first update the estimate using the estimate from the previous iteration and then combine the updates from all neighbors to form the final estimate for the iteration. This method is known as Adapt-then-Combine (ATC) diffusion. Simulation results show that ATC diffusion outperforms CTA diffusion.
Using LMS, the CTA diffusion algorithm is given by:
                              {                                                                                          ψ                                          k                      ,                      i                                                        =                                                            φ                                              k                        ,                        i                                                              +                                                                  μ                        k                                            ⁢                                                                        u                                                      k                            ,                            i                                                    *                                                ⁡                                                  (                                                                                                                    d                                k                                                            ⁡                                                              (                                i                                )                                                                                      -                                                                                                                            u                                  k                                                                ⁡                                                                  (                                  i                                  )                                                                                            ⁢                                                              φ                                                                  k                                  ,                                  i                                                                                                                                              )                                                                                                                                                                                                              φ                                          k                      ,                      i                                                        =                                                            ∑                                              l                        ∈                                                  N                          k                                                                                      ⁢                                                                  c                        lk                                            ⁢                                              ψ                                                  l                          ,                                                      i                            -                            1                                                                                                                                                                            }                ,                            (        2        )            where {clk}IεNk is a combination weight for node k, which is fixed, φk,i-1 is the combined estimate, and ψk,i is the estimate for node k at iteration i, and μk is the node step size.
The ATC algorithm is achieved by reversing the order of the equations of equation set (2):
                              {                                                                                          φ                                          k                      ,                      i                                                        =                                                            ψ                                              k                        ,                                                  i                          -                          1                                                                                      +                                                                  μ                        k                                            ⁢                                                                        u                                                      k                            ,                            i                                                    *                                                ⁡                                                  (                                                                                                                    d                                k                                                            ⁡                                                              (                                i                                )                                                                                      -                                                                                                                            u                                  k                                                                ⁡                                                                  (                                  i                                  )                                                                                            ⁢                                                              ψ                                                                  k                                  ,                                                                      i                                    -                                    1                                                                                                                                                                                )                                                                                                                                                                                                              ψ                                          k                      ,                      i                                                        =                                                            ∑                                              l                        ∈                                                  N                          k                                                                                      ⁢                                                                  c                        lk                                            ⁢                                              φ                                                  l                          ,                          i                                                                                                                                                  }                .                            (        3        )            
FIG. 3 diagrammatically illustrates the ATC algorithm implemented in an adaptive network of N nodes. The objective for a diffusion algorithm is to minimize the cost function J(w) with respect to unknown vector w, where J(w) is given by:
                              min          ⁢                                          ⁢                      wJ            ⁡                          (              w              )                                      =                              ∑                          k              =              1                        N                    ⁢                                    E              ⁡                              [                                                                                                                        d                        k                                            -                                                                        u                          k                                                ⁢                        w                                                                                                  2                                ]                                      .                                              (        4        )            
Equation (4) is a global cost function. The local cost function at each node is given by:Jk(w)=E└|dk−ukw|2┘.  (5)Completing the squares and using the notation E[uk*uk]=Ru,k, equation (5) can be rewritten as:Jk(w)=∥w−wk∥Ru,k2+mmse.  (6)The global cost function now becomes:
                              J          ⁡                      (            w            )                          =                              E            ⁡                          [                                                                                                            d                      k                                        -                                                                  u                        k                                            ⁢                      w                                                                                        2                            ]                                +                                    ∑                              l                ≠                k                            N                        ⁢                                                                                                  w                    -                                          w                      l                                                                                                          R                                      u                    ,                    l                                                  2                            .                                                          (        7        )            
The above model assumes that each node has access to the entire network. In a practical system, however, a node has access only to its closest neighbors. Thus, the cost function has to be approximated with only data from neighbors being shared at each node. As a result, the weighting matrix for the second term does not remain Ru,l but, instead, has to be replaced by a constant weighting factor blk. The value of wl is also replaced by its intermediate estimate from node l, i.e., ψl. Eventually, the cost function becomes:
                                          J            k                    ⁡                      (            w            )                          =                              E            ⁢                          ⌊                                                                                                            d                      k                                        -                                                                  u                        k                                            ⁢                      w                                                                                        2                            ⌋                                +                                    ∑                              l                ∈                                                      N                    k                                    ⁢                  l                  ⁢                                      {                    k                    }                                                                        ⁢                                          b                lk                            ⁢                                                                                                              w                      -                                              ψ                        l                                                                                                  2                                .                                                                        (        8        )            Defining Jk1(w)=E└|dk−ukw|2┘ yields:
                              min          ⁢                                          ⁢                                    wJ              k                        ⁡                          (              w              )                                      =                                            J              k              1                        ⁡                          (              w              )                                +                                    ∑                              l                ∈                                                      N                    k                                    ⁢                  l                  ⁢                                      {                    k                    }                                                                        ⁢                                          b                lk                            ⁢                                                                                                              w                      -                                              ψ                        l                                                                                                  2                                .                                                                        (        9        )            
Conventional LMS algorithms are not equipped to efficiently provide such estimates, particularly in complex and large adaptive networks. The diffusion scheme based on conventional LMS algorithms lacks knowledge of noise power in the system. Thus, it cannot perform well in the presence of noise. Thus, a noise-constrained diffusion least mean square method for estimation in adaptive networks solving the aforementioned problems is desired.